Discrete lattice effects and the phason gap of incommensurate systems

The effect of thermal fluctuations and discreteness of the crystal lattice on the pinning of the modulation wave and the phason gap of structurally incommensurate systems is examined. Very close to the paraelectric-to-incommensurate transition temperature TI, the phason gap is vanishingly small, but at slightly lower temperatures it takes on a nonzero value. Near the lower incommensurate-to-commensurate transition temperature TC, where floating effects are negligible, the pinning may lead to a ``devil's staircase''-like temperature dependence of the phason gap.